openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Entropy /CurveRationalLaw.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Numerical.CurveLawsCertificates | |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ConditionalContinuity | |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ConditionalRateBridge | |
| namespace OAI | |
| /-! Finite entropy, rate estimates and ordered asymptotic limits. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CurveLaws | |
| open MatrixMultiplication.Foundation ComplexWitness ConditionalLabels Filter | |
| open scoped BigOperators Topology | |
| theorem exists_rational_rate_bounds : | |
| ∃ q : RationalLaw CurveLeaf, | |
| (31519 : ℝ) / 10000 < | |
| lawPairingRate q.toFiniteLaw curveLabels 5 curveReaderPair .xy + | |
| lawPairingRate q.toFiniteLaw curveLabels 5 curveReaderPair .xz ∧ | |
| lawPairingRate q.toFiniteLaw curveLabels 5 curveReaderPair .xy + | |
| lawPairingRate q.toFiniteLaw curveLabels 5 curveReaderPair .xz < | |
| (15761 : ℝ) / 5000 ∧ | |
| (27961 : ℝ) / 25000 < | |
| lawPairingRate q.toFiniteLaw curveLabels 5 curveReaderPair .yz ∧ | |
| lawPairingRate q.toFiniteLaw curveLabels 5 curveReaderPair .yz < | |
| (6991 : ℝ) / 6250 := by | |
| obtain ⟨q, hq⟩ := exists_rational_law_sequence (leafLaw (2 / 3 : ℝ)) | |
| have hi := (tendsto_lawPairingRate hq curveLabels 5 curveReaderPair .xy).add | |
| (tendsto_lawPairingRate hq curveLabels 5 curveReaderPair .xz) | |
| have hh := tendsto_lawPairingRate hq curveLabels 5 curveReaderPair .yz | |
| rw [incident_rate_two_thirds] at hi | |
| rw [hidden_rate_two_thirds] at hh | |
| have hilo := hi.eventually_const_lt ComplexCertificates.curveH_bounds.1 | |
| have hihi := hi.eventually_lt_const ComplexCertificates.curveH_bounds.2 | |
| have hhlo := hh.eventually_const_lt ComplexCertificates.curveG_bounds.1 | |
| have hhhi := hh.eventually_lt_const ComplexCertificates.curveG_bounds.2 | |
| obtain ⟨n, hn⟩ := (hilo.and (hihi.and (hhlo.and hhhi))).exists | |
| exact ⟨q n, hn⟩ | |
| theorem exists_count_rate_bounds : | |
| ∃ counts : CurveLeaf → ℕ, ∃ hD : 0 < ∑ a, counts a, | |
| (31519 : ℝ) / 10000 < | |
| lawPairingRate (normalizedCountLaw counts hD) curveLabels 5 curveReaderPair .xy + | |
| lawPairingRate (normalizedCountLaw counts hD) curveLabels 5 curveReaderPair .xz ∧ | |
| lawPairingRate (normalizedCountLaw counts hD) curveLabels 5 curveReaderPair .xy + | |
| lawPairingRate (normalizedCountLaw counts hD) curveLabels 5 curveReaderPair .xz < | |
| (15761 : ℝ) / 5000 ∧ | |
| (27961 : ℝ) / 25000 < | |
| lawPairingRate (normalizedCountLaw counts hD) curveLabels 5 curveReaderPair .yz ∧ | |
| lawPairingRate (normalizedCountLaw counts hD) curveLabels 5 curveReaderPair .yz < | |
| (6991 : ℝ) / 6250 := by | |
| obtain ⟨q, hq⟩ := exists_rational_rate_bounds | |
| obtain ⟨counts, hD, hmass⟩ := q.exists_type_representation | |
| have law_eq (p r : FiniteLaw CurveLeaf) (h : p.mass = r.mass) : p = r := by | |
| cases p | |
| cases r | |
| cases h | |
| rfl | |
| have heq : normalizedCountLaw counts hD = q.toFiniteLaw := | |
| law_eq _ _ (funext hmass) | |
| refine ⟨counts, hD, ?_⟩ | |
| simpa only [heq] using hq | |
| end MatrixMultiplication.CurveLaws | |
| end | |
| end OAI | |